"Kill 5 enemy crew in a fight without taking hull damage or losing a crew member while using the Mantis Cruiser."
Besides a fight against an enemy ship, this achievement can be won from a shipless boarding event with five or more attackers. Boarding and Anti-Personnel drones can be used to help unlock this achievement. Otherwise, many Rebel ships with crews of five or more are easily found in The Last Stand sector, provided you can get through a battle there without taking hull damage, which is the real catch.
With Advanced Edition's Clone Bay system, this achievement is made massively easier: Find a ship with the system installed, and after killing each crewmember, simply allow them to finish cloning, and kill them again until you've hit 5 kills. The entire enemy crew can be dead at once without finishing the battle, provided their Clone Bay is still online and powered. Ships with pre-installed Clone Bays can be found anywhere, although they seem to spawn less frequently than ships without them.
An oversight allows collecting of the achievement even after taking damage during the fight if it is recovered afterwards with a Hull Repair Drone before jumping.